Automorphism groups of compact bordered Klein surfaces. A combinatorial approach
type: Item
instance of: scholarly article
title: Automorphism groups of compact bordered Klein surfaces. A combinatorial approach
zbMATH Open document ID: 0709.14021
author: G. Gromadzki
author: Emilio Bujalance
author: José Javier Etayo
author: Jose Manuel Gamboa
published in: Lecture Notes in Mathematics
publication date: 1992-09-17T00:00:00Z
review text: In complex algebraic geometry, the study of the birational (or holomorphic) automorphism groups of algebraic curves (i.e., compact Riemann surfaces) has a long history, which can be traced back to the pioneering work of H. A. Schwarz, A. Hurwitz, F. Klein, P. Gordan, and A. Wiman at the end of the last century. Although the problem of determining all possible (finite) groups of automorphisms of a given Riemann surface of genus \( g\) appears very difficult, in general, and still far from being completely solved, a great progress has been achieved in the recent decades, basically due to the contributions of A. M. Macbeath, R. Accola, C. Machlachlan, H. Farkas, R. Kulkarni, and many others. The main tools for the study of automorphisms groups of Riemann surfaces are, in particular, the theory of Fuchsian groups (representing Riemann surfaces by uniformization) and the theory of Weierstrass points. In real algebraic geometry, the study of birational automorphism groups of real curves has begun much later, actually as late as about 20 years ago. An essential step forward had been made by the paper of \textit{N. L. Alling} and \textit{N. Greenleaf} [cf.: ``Foundations of the theory of Klein surfaces'', Lect. Notes Math. 219 (1971; Zbl 0225.30001)], in which the so-called Klein surfaces were introduced. These Klein surfaces (i.e., real surfaces admitting a dianalytic structure) turned out to be the right objects for extending the study of automorphisms of Riemann surfaces to the case of the real groundfield, in that they have real function fields in one variable as fields of ``meromorphic functions'' and, therefore, are categorically equivalent to real algebraic curves. On the other hand, by a fundamental result of \textit{R. Preston} [cf. ``Projective structures and fundamental domains on compact Klein surfaces'', Ph. D. Thesis (Univ. Texas, 1975); unpublished], there is an analogue of Poincaré's uniformization theorem for Klein surfaces, which states that a Klein surfac…
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 30F35
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 30F50
Mathematics Subject Classification ID: 14P15
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 30-02
Mathematics Subject Classification ID: 20-02
zbMATH DE Number: 47697
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: Weierstrass points
zbMATH Keywords: real algebraic geometry
zbMATH Keywords: birational automorphism groups of real curves
zbMATH Keywords: Siegel modular group
zbMATH Keywords: NEC groups
zbMATH Keywords: Teichmüller spaces
zbMATH Keywords: uniformization theory for Klein surfaces
zbMATH Keywords: automorphism groups of real algebraic curves
reviewed by: Werner Kleinert
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bfb0084977
OpenAlex ID: W4252932738
DOI: 10.1007/BFB0084977
description: scientific article; zbMATH DE number 47697